Question: Solve for $x$ : $ 4|x - 5| + 1 = 6|x - 5| + 6 $
Solution: Subtract $ {4|x - 5|} $ from both sides: $ \begin{eqnarray} 4|x - 5| + 1 &=& 6|x - 5| + 6 \\ \\ {- 4|x - 5|} && {- 4|x - 5|} \\ \\ 1 &=& 2|x - 5| + 6 \end{eqnarray} $ Subtract $6$ from both sides: $ \begin{eqnarray} 1 &=& 2|x - 5| + 6 \\ \\ {- 6} && {- 6} \\ \\ -5 &=& 2|x - 5| \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{-5} {{2}} = \dfrac{2|x - 5|} {{2}} $ Simplify: $ -\dfrac{5}{2} = |x - 5| $ The absolute value cannot be negative. Therefore, there is no solution.